Step-strain experiment

It became clear in the early development of the tube model that it provided a means of calculating the response of entangled polymers to large deformations as well as small ones [2]. Some predictions, especially in steady shear flow, lead to strange anomaUes as we shall see, but others met with surprising success. In particular the same step-strain experiment used to determine G(t) directly in shear is straightforward to extend to large shear strains y. In many cases of such experiments on polymer melts both Hnear and branched, monodisperse and polydisperse,the experimental strain-dependent relaxation function G(t,Y) may be written... 

The step-strain experiments discussed above furnish the simplest example of a strong flow. Many other flows are of experimental importance transient and steady shear, transient extensional flow and reversing step strains, to give a few examples. Indeed the development of phenomenological constitutive equations to systematise the wealth of behaviour of polymeric liquids in general flows has been something of an industry over the past 40 years [9]. It is important to note that it is not possible to derive a constitutive equation from the tube model in... 

The relaxation modulus G(t) is the value of the transient stress per unit strain in a step-strain experiment. This type of experiment may be achieved with modem rotary rheometers with a limited resolution in time (roughly 10 2 s). If one wishes to evaluate G(t) at shorter times, it is necessary to derive G(t) from the high frequency G (co) data by an inverse Carson-Laplace transform. 

Two additional factors related to constraint release need to be considered for finite step strain experiments in liquids ... 

Stress relaxation in step strain experiments on a viscoelastic solid (upper curve) and a viscoelastic liquid (lower curve). The dashed lines show the value of the stress at the relaxation time t of the liquid. The solid has the same relaxation time. 

For viscoelastic liquids, the Maxwell model can be used to qualitatively understand the stress relaxation modulus. In the step strain experiment, the total strain 7 is constant and Eqs (7.101)-(7.103) can be combined to give a first order differential equation for the time-dependent strain in the viscous element ... 

In practice, it is more precise to evaluate the viscosity from a step strain experiment by transforming the integration of Eq. (7.117) to a logarithmic time scale using the identity td nt = dt, and the lower limit of integration changes because when = 0, In f = - 00 ... 

The improved precision of Eq. (7.121) as compared to Eq. (7.117), for determining the viscosity from the step strain experiment, arises from the fact that tG t) is a function with a well-defined peak, and the relaxation modulus can decay over many decades of time for viscoelastic materials, such as polymers. 

Thus far we have imposed a constant strain (the step strain experiment) and constant shear rate (the steady shear experiment). Another simple... 

Under such hypothesis, for the step-strain experiment of Fig. 1, the evolution of one chain in the sample is schematically presented in Fig. 5 immediately after the deformation the chain is elongated in its... 

Constant stress is often a natural loading Natural way to do step strain experiments. 

The total displacement, X is simply + Xj, and for a step-strain experiment is given by a step function. The differential equation can then be solved as follows for F(t) ... 

To develop the tube theory of polymer motion, we consider the response of the melt to a step deformation. This is an idealized deformation that is so rapid that during the step no polymer relaxation can occur, and the polymer is forced to deform affinely, that is, to the same degree as the macroscopic sample is deformed. The total deformation, though rapid, is small, so that the chains deform only slightly this is called a small amplitude step strain. Because the deformation is very small, the distribution of chain configurations remains nearly Gaussian, and linear viscoelastic behavior is expected. In Chapter 4 we saw that the assumption of linear behavior makes it possible to use the response to a small step strain experiment to calculate the response to oscillatory shear or any other prescribed deformation. 

For a cured rubber, there is a unique configuration of a material element that it will always return to when the extra stress is zero, and a time when the element was in this configuration is an obvious choice for the reference time. For a melt, there is no such unique, unstrained state, so some other reference time must be selected. In a laboratory experiment in which a sample of a melt is initially in a fixed, stress-free configuration, the time at which the deformation begins is an obvious reference time. For example, for a step strain experiment, the relaxation modulus G(f) is measured as a function of the time from the instant of the initial strain (t = 0). Thus it is convenient to let the reference time be fg = 0. 

It was pointed out in Section 10.4.3 that wall slip can cause a large error in the determination of the strain in step-strain experiments, and the true strain maybe much less than the nominal strain inferred from the displacement of a rheometer surface. The observation that N /a is independent of time does not, by itself, imply that there is no slip unless this ratio is also equal to the nominal strain applied. And when the Lodge-Meissner rule is not obeyed, it is often taken as evidence that slip is occurring, and the stress ratio Nj/cris used in place of the nominal strain as the independent variable in reporting shear stress and normal stress difference data [40]. 

Controlled strain is the preferred mode of operation for nonlinear studies. In step-strain experiments, an important source of experimental error is the deviation of the actual strain history from a perfect step. Laun [96] and Venerus and Kahvand [43] have discussed this problem and how it can be addressed. Gevgilili and Kalyon [ 100] found that the actual strain pattern generated by a popular coimnerdal rheometer in response to a command for a step was, in fact, a rather complex function of time. One approach that is of use in comparing data from any transient test with the predictions of a model is to record the actual, non-ideal, strain history and use this same history to calculate the model predictions. 

Step strain experiments have been carried out using lubricated squeeze flow to determine the damping function for biaxial extension h e ). The Doi-Edwards tube model (without lA assumption) prediction of this function is as follows [24] ... 

Vrentas, C. M., Graessley, W. W. Relaxation of shear stress and normal stress components in step-strain experiments J. Non-Newt FI. Mech. (1981) 9, pp. 339-335... 

Soskey, P. R., Winter, H. H. Large step strain experiments with parallel-disk rotational rheometers. /. Rheol. (1984) 28, pp. 625-645... 

Figures 6a-b show the extensional viscosity of the branched samples at 180°C. The measurements were conducted at different extension rates and compared to the linear viscoelastic (LVE) envelope determined by a step strain experiment at the same temperature. The two samples show strain hardening characterized by a deviation of the transient viscosity from the LVE envelope. The deviation at all extension rates confirms the presence of chain branching but more importantly shows a relationship between the extension rate and the extent of hardening. This correlation is considered crucial during parison inflation as it allows better control during the inflation of the parison.

Extensional viscosity measurements were determined at 180°C and different extension rates using an Exteitsional Viscosity Fixture (EVF) irrstalled in the ARES-LS rheometer. The linear viseoelastie (LVE) envelope was determined by a step-strain experiment at low strain rates and converted to the zero-exterrsion rate based on the Trouton s rule. 

Another method to investigate the effect of chain branching on the rheology of the samples is by measuring the relaxation modulus, which is typically done using a step strain experiment. The relaxation modulus was calculated using the Ninomiya-Ferry approximation ... 

---